the contribution of horizontal arching to tunnel face stability 2012 geotechnik

34 2012 Ernst & Sohn Verlag fr Architektur und technische Wissenschaften GmbH & Co. KG, Berlin geotechnik 35 (2012), Heft 1

Fachthemen

DOI: 10.1002/gete.201100024

The article revisits the classic problem of tunnel face stabilitywith special emphasis on the effect of horizontal stresses. Theseare important for shear resistance and thus also for the equilibri-um of the potentially unstable body in front of the tunnel face, butthey also present the difficulty of static indeterminacy. Startingfrom the computational model of Anagnostou and Kovri [1], analternative model is presented, which is based on the so-calledmethod of slices, and is consistent with silo theory, but does notneed an a priori assumption as to the distribution of horizontalstress. In addition, a simple design equation for estimating sup-port pressure under this model is presented and the results ofcomparative analyses concerning the average stresses in thewedge and the effects of shear resistance at the lateral slip sur-faces are shown. The analytical results obtained by the methodof slices agree very well with published results of numericalanalyses and physical tests.

Beitrag der rumlichen Tragwirkung zur Stabilitt der Tunnel-brust. Der vorliegende Artikel untersucht das klassische Problemder Stabilitt der Ortsbrust unter besonderer Beachtung der Hori-zontalspannungen. Letztere sind zwar sehr wichtig fr den Gleit -widerstand und somit auch fr die Stabilitt von potenziellenBruchkrpern, knnen aber nicht allein aufgrund von Gleichge-wichtsbetrachtungen ermittelt werden. Im Beitrag wird eine Berechnungsmethode vorgestellt, die das Berechnungsmodellnach Anagnostou und Kovri [1] insofern verbessert, dass sie keine a priori Annahme ber die Verteilung der Spannungen imkeilfrmigen Bruchkrper vor der Ortsbrust bentigt und auf konsistente Weise das Gleichgewicht im Keil und im darber liegenden prismatischen Bruchkrper analysiert. Basierend aufder Lamellenmethode wird eine einfache Bemessungsformel auf-gestellt und der Einfluss der horizontalen Verspannung auf denerforderlichen Sttzdruck der Ortsbrust aufgezeigt. Die Modell-prognosen stimmen mit verffentlichten Ergebnissen von numerischen Spannungsanalysen sowie mit Versuchsresultatengut berein.

1 Introduction

In contrast to long excavations, where the relevant shearstresses are mobilized only at the inclined slip surfaceand the stability problem is practically two-dimensional(Figure1a), the load bearing action of the ground ahead ofthe tunnel face is three-dimensional. This can best be il-lustrated by considering the failure model of a potentiallyunstable wedge at the face (Figure1b): The shear stressess developing at the two vertical slip surfaces contribute to

the stability of the wedge; the term horizontal arching canjustifiably be used in this context because the direction ofthe shear and normal stresses acting upon the lateralboundaries of the wedge show that the principal stress tra-jectories must be oriented as indicated by the dashed linesin Figure 1b.

There are many publications dealing with theoreticaland experimental investigations into tunnel face stability.Recent reviews may be found, for example, in Idinger et al.[2], Mollon et al. [3] and Perazzelli and Anagnostou [4].The present paper analyses the contribution of horizontalarching to stability on the basis of the computational mod-el of Anagnostou and Kovri [1], which was developed inthe context of slurry shield tunnelling and is widely usedin engineering practice. The model approximates the tun-nel face by a rectangle (of height H and width B) and con-siders a failure mechanism that consists of a wedge at theface and an overlying prism up to the soil surface (depthof cover h, Figure2).

The contribution of horizontal arching to tunnel face stability

Georgios Anagnostou

Fig. 1. a) Cross section and horizontal plan of a long pit under plane strain conditions; b) longitudinal and horizontalsection of a tunnelBild 1. a) Querschnitt und Grundriss einer langen Baugrubeim ebenen Verformungszustand, b) Lngsschnitt und Hori-zontalschnitt eines Tunnels

35

G. Anagnostou The contribution of horizontal arching to tunnel face stability

geotechnik 35 (2012), Heft 1

The central problem of horizontal arching is associ-ated with the estimation of shear resistance at the verticalslip surfaces of the wedge (s in Figure1): The frictionalpart of the shear resistance depends on the horizontalstress y, which nevertheless cannot be derived from theequilibrium conditions (it is statically indeterminate). Thisproblem is due to the spatial geometry of the failure mech-anism and it also arises in stability analyses of slurry walltrenches or excavations with large depth to width ratios.In order to overcome this difficulty in the analysis of deepexcavations, Walz and Pulsfort [5] assumed, (i), that thehorizontal stress y (which governs the frictional resis-

tance at the vertical slip surfaces) is linearly proportionalto the vertical stresses z, i.e.

(1)

where is a constant (the so-called lateral stress coeffi-cient), and, (ii), that the vertical stress z changes linearlywith depth. This assumption was also made in the Ger-man specifications for slurry wall design [6] and was madeby Anagnostou and Kovri [1] in their computationalmodel:

(2)

where denotes the unit weight of the soil. The stressz(H) at the top boundary of the wedge is obtained by ap-plying silo theory to the overlying prism. The solid line inFigure 3 represents the stress distribution under this as-sumption, while the dashed lines show alternative formu-lations discussed by Broere [7]: Line 1 disregards archingeffects in respect of the wedge, line 2 assumes that archingin respect of the wedge can also be approximated by theclassic silo equation (in spite of its non-constant horizon-tal cross-section) and line 3 represents a compromise be-tween model 1 and 2.

The advantage of all these approaches is their sim-plicity. The disadvantage, however, is the a priori nature ofthe assumption concerning the vertical stress z and thelack of consistency regarding the analysis of the prismaticbody, which faces exactly the same problem, but solves itin a different way, i.e. on the basis of Janssens silo theory[8].

A more consistent way of calculating the frictionalpart of the shear resistance at the lateral slip surfaces ofthe wedge is to proceed by analogy with silo theory, i.e. tokeep the assumption of proportionality between horizon-tal and vertical stress (Equation 1), but, in order to calcu-late the distribution of the vertical stresses z inside thewedge, to consider the equilibrium of an infinitesimallythin slice (Figure 4). Walz and Prager [9] first proposedsuch an approach for the stability assessment of slurrywalls. This so-called method of slices eliminates the needfor an a priori assumption as to the distribution of the ver-tical stress and makes it possible to analyse cases withnon-uniform face support and heterogeneous ground con-sisting of horizontal layers. The method of slices alsomakes it possible to estimate on a more consistent basis(similarly to silo theory) the vertical stresses within thewedge. It should be noted that the stresses (y, z) withinthe wedge are important not only with respect to the fric-tional resistance at the vertical slip surfaces but also withrespect to the pull-out resistance of the bolts which may beinstalled in order to stabilise the face (a high confiningstress increases the strength of the bond between boltsand soil).

The paper in hand analyses tunnel face stability us-ing the method of slices (Section 2), discusses the resultsof comparative analyses concerning the stresses in thewedge (Section 3) and their effects on the required sup-port pressure (Sections 4 to 6), and proposes a simple de-sign equation (Section 7).

(z) (H) zH

H 1 zHz z

= +

y z =

Fig. 3. Assumption of Anagnostou and Kovri [1] concerningthe vertical stress distribution (solid line) as well as alterna-tive formulations discussed by Broere [7] (dashed lines)Bild 3. Annahme von Anagnostou und Kovri [1] ber dieVerteilung der Vertikalspannung (durchzogene Linie) sowiealternative Annahmen nach Broere [7] (gestrichelte Linien)

Fig. 4. Forces acting upon an infinitesimal sliceBild 4. Krfte auf einer infinitesimalen Lamelle

Fig. 2. Failure mechanismBild 2. Bruchmechanismus

36



2 Computational model2.1 Outline

In the mechanism under consideration (see Figure2), fail-ure will occur if the load exerted by the prism upon thewedge exceeds the force which can be sustained by thewedge at its upper boundary taking into account the shearstrength and the own weight of the ground. At limit equi-librium the prism load is equal to the bearing capacity ofthe wedge. The prism load is calculated on the basis of si-lo theory (Section 2.2), while the bearing capacity of thewedge is calculated by considering the equilibrium of aninfinitesimal slice (Section 2.3). Both the load of the prismand the bearing capacity of wedge depend on the inclina-tion of the inclined slip plane. The critical value of the an-gle (see Figure2), i.e. the value that maximizes the sup-port pressure, will be determined iteratively.

2.2 Prism loading

Assuming that the ground is homogeneous and obeys theMohr-Coulomb failure condition with cohesion c and an-gle of internal friction , the vertical force at the wedge-prism interface reads as follows:

(3)

where surf denotes the surface load and R is equal to theratio of the volume of the prism to its circumferentialarea:

(4)

Equation (3) ensures that the load exerted by the prismwill be set equal to zero (rather than becoming negative) ifthe cohesion exceeds the critical value:

(5)

2.2 Bearing capacity of the wedge

Consider the equilibrium of an infinitesimal slice (see Fig-ure4). In the plane of movement, the following forces actupon the slice: Its weight dG; the supporting force V(z)exerted by the underlying ground; the loading force V(z)+ dV exerted by the overlying ground; the forces dN anddT at the inclined slip surface; the shear force dTs at thetwo vertical slip surfaces; and the supporting force dS.The equilibrium conditions parallel and perpendicular tothe sliding direction read as follows:

(6)

(7)

The slice weight is

(8)

V min 0, R ctan

1 e e

BHtan

silotan h

Rsurf

tan hR=

+

R BHtan2 B Htan( )

= +

c R (if 0)cr surf= =

dT dT dSsin (dV dG)coss + + = +

dN (dV dG)sin dScos= + +

dG B dA=

where dA denotes the area of the lateral boundary of theinfinitesimal slice:

(9).

The support force is

(10)

where s denotes the support pressure.According to the Mohr-Coulomb criterion, the shear

resistance dT of the inclined slip surface is connected tothe normal force dN:

(11)

The shear resistance of the two lateral slip surfaces readsas follows:

(12)

where z represents the horizontal normal stress (Equa-tion 1). Taking into account equation (9) and that the ver-tical stress

(13)

we obtain:

(14)

Due to equations (7), (8), (10), (11) and (14), the equilibri-um condition in the sliding direction (Equation 6) be-comes:

(15)

where

(16)

(17)

(18)

(19)

(20)

(21)

(22)

Equation (15) is a differential equation for the verticalforce V(z). Assuming a homogeneous ground and uniformsupport pressure distribution, the coefficients , and do not depend on the co-ordinate z and the solution to

dA= z tan dz

dS s B dz=

dT B dzcos

c + dNtan=

dT 2 dA c + tans z( )=

VBztanz

=

dT 2 c tan z dz 2 tan VB

dzs = +

B dVdz

V M zB

+ P =

2 tancos sin tan

=

M = M B c M Bc2 3

P = P B c + P B sc2

s2

M tantanc

=

M tan=

P2 tan cosc

=

P tans ( )= +

37



equation (15) for the boundary condition V(0) = 0 reads asfollows:

(23)

where is the normalized z co-ordinate,

(24)

and Cs, Cc and C are dimensionless functions of :

(25)

(26)

(27)

(28)

(29)

The bearing capacity of the wedge is obtained from equa-tion (23) with z = H:

(30)

The coefficients Cs, Cc and C express the effect of supportpressure, cohesion and unit weight, respectively, on thebearing capacity of the wedge.

2.4 Support pressure

At limit equilibrium the load exerted by the prism is equalto the bearing capacity of the wedge:

(31)

As V(H) depends linearly on s (Equation 30), equation(31) represents a linear equation for the support pressure s.Its solution reads as follows:

(32)

where

(33)

(34)

, (35)

and, according to equation (3),

V(z) C ( )B s C ( )B c C ( )Bs2

c2 3= +

zH

=

C ( ) evH/B = ( )

C ( )C ( ) 1

Psv

s =

C ( )C ( ) 1

PF ( )

Mcv

c 2 c =

+

F( ) C ( ) 1 HBv

=

V(H) C (1)B s C (1)B c C (1)Bs2

c2 3= +

V(H) V!

silo=

sH

f f cH

f(H)H1 2 3

z

=

+

f BH

C (1)

C (1)f B

H, , ,1

s1= =

fC (1)C (1)

f BH

, , ,2c

s2= =

C ( ) F( ) M2

=

f tanBH

C (1)f B

H, , ,3

s

3= =

, (36)

It can easily be verified that with increasing depth of cov-er h the exponential term in equation (3) decreases rapid-ly to zero with the consequence that the silo pressure andthe necessary face support pressure become practically in-dependent of the depth of cover (quantitative examplesare given in Section 4). With the exception of very shallowtunnels, and provided that the cohesion is lower than R(i.e. that the prism needs support in order to be stable),both the expression (36) for the silo loading and the ex-pression (32) for the support pressure become consider-ably simpler for large values of h:

(37)

Where

(38)

and

(39)

where

(40)

(41)

It can readily be verified that for = 1 equation (41) sim-plifies to:

(42)

It is remarkable that this result is identical to the numeri-cal results by Vermeer and Ruse [10] and Vermeer et al. [11](note that f52 = ds/dc). As mentioned by Ruse [12], the re-lationship (42) is theoretically founded and close to theequation ds/dc = 0.5 cot proposed by Krause [13] on thebasis of a completely different failure mechanism (the slid-ing of a semi-spherical body at the face).

2.5 Distribution of the vertical stress

From equations (13), (23) and (32) we obtain the averagevertical stress z of the wedge slice at elevation z:

(H)H

f RH

f cH

z81 81

=

f 1tan81

=

sH

f f cH51 52

=

f f f f RH

f BH

, , ,51 1 3 81 51= + =

f f f f f BH

, , ,52 2 3 81 52= + =

f 1tan52

=

(H)H

1H

VBH tan

min 0,

RH

cH

tan1 e

He

f BH

, , , , hH

, cH

,H

z silo

tan hR surf

tan hR

7surf

=

=

+

=

38



(43)

where

(44)

(45)

(46)

At the wedge foot, the nominators and the denominatorsof these equations become equal to zero. The stress can becomputed by applying LHpitals rule:

(47)

2.6 Frictional resistance of the vertical slip surfaces

As an overall measure for the frictional resistance, the av-erage frictional stress av, may be considered. Accordingto Coulomb and equation (1),

(48)

where z,av is the average vertical stress. The latter can becalculated via integration over the lateral wedge bound-ary:

(49)

where

(50)

(51)

(52)

(53)

For comparison, the average vertical stress in the case of alinear distribution according to equation (2) reads as fol-lows:

tanav, z,av =

(0) 1B tan

lim dV(z)dz

P c P stanz z 0

c s =

=+

f 1C ( )C (1)11

s

s

=

fB / H

tanf C (1)

C ( )12

2

11( )

=

f B / Htan

f C (1)C ( )

13 11 cc=

(z) BH

C ( )s C ( )c C ( )B

tan

f (H) f H f c

zs c

11 z 12 13

= +

=

= +

z tan dz

z tan dz

2H

zdz

f s f c f H

z,av

z0

H

0

H 2 z0

H

21 22 23

=

=

= +

ftan

tanf21 25

( )=

+

f 1tan

f2 sin

f 12225

25

=

+

f BH

f 123 25( )=

f 2 BH

BH

e 1 125H/B( )=

. (54)

Note that, although the stress distribution of equation (2)is linear, the average vertical stress according to equation(54) corresponds to the stress prevailing at elevation z =2H/3 rather than to the stress at the tunnel axis. This isdue to the larger contribution of the upper part of thewedge.

3 Comparative calculations concerning stress distribution

The linear approximation under equation (2) has been ex-amined by Walz and Pulsfort [5] in the context of slurrywall stability. In this paper, the results of comparative cal-culations for the problem of tunnel face stability will bediscussed.

Consider a tunnel with width B = 10 m, heightH= 10m and cover h > H in a homogeneous ground with = 20kN/m3. Figure 5 shows the vertical stress z overthe face height z obtained using the method of slices(Equation 43, solid lines) or assuming the linear distribu-tion of equation (2) (dashed lines) for three sets of shearstrength parameters: cohesionless soil with = 25 or 35and cohesive soil with = 25 and c = 20 kPa. In bothmodels, the stress at the upper boundary of the wedge isequal to the silo pressure. The latter was calculated as-suming the coefficient of lateral stress = 0.8. This valueis supported by the results of trap-door tests by Melix [14],which indicate that is between 0.8 and 1.0, which isslightly lower than the value of 1 suggested by Terzaghiand Jelinek [15]. The computation of the vertical stress zusing the method of slices also necessitates an assumptionconcerning the coefficient for the wedge. On account of

23

H 13

Hz,av,lin z ( ) = +

Fig. 5. Distribution of vertical stress z over the height ofthe tunnel face for a wedge with = 30 (other parameters: = 20 kN/m3, B = H = 10 m, h > H, = 0.80)Bild 5. Verteilung der Vertikalspannung z ber die Orts-brusthhe fr einen Keil mit = 30 (sonstige Parameter: = 20 kN/m3, B = H = 10 m, h > H, = 0,80)

39



the similarity of this model to the silo theory, the same val-ue of = 0.8 was assumed for the wedge and the prism.

According to the equations in Section 2.5, the verti-cal stress distribution depends essentially on the angle ,while this parameter has a minor effect when assuming thesimplified linear distribution of equation (2) (it affects on-ly the silo load Vsilo). Figure 5 was obtained for = 30. In

addition to each line, the diagram also shows the value ofthe average vertical stress (calculated on the basis of Eqs.49 and 54). It can easily be seen that the assumption ofequation (2) leads to vertical stresses that are considerablyhigher than the stresses obtained using the method ofslices. This is particularly true in the case of the higherstrength soils ( = 35 or c = 20 kPa), because the lateralshear resistance does not allow the stress to increase withdepth in the method of slices, while the linear stress distri-bution of equation (2) does not explicitly consider theshear strength of the ground.

Due to the higher vertical stress in the simplifiedmodel, the lateral frictional resistance will also be higherthan with the method of slices. The simplified model thuspredicts a lower support pressure. This is clearly illustrat-ed by the diagrams of Figure 6, which present the neces-

Fig. 7. a) Ratio of the average vertical stresses z,av/z,av,linand, b), necessary support pressure s according to the me-thod of slices as a function of cohesion c for diferent valuesof the friction angle and of the angle (other parameters: = 20 kN/m3, B = H = 10 m, h > H, = 0.80)Bild 7. a) Verhltnis der mittleren Vertikalspannungenz,av/z,av,lin und, b), erforderlicher Sttzdruck s nach der Lamellenmethode in Abhngigkeit der Kohsion c fr ver-schiedene Werte des Reibungwinkels und des Winkels (sonstige Parameter: = 20 kN/m3, B = H = 10 m, h > H, = 0,80)

Fig. 6. Support pressure s as a function of the angle for the parameters of Figure5 and (a) = 25 and c = 0; (b) = 35 and c = 0 kPa; (c) = 25 and c = 20 kPaBild 6. Sttzdruck s in Abhngigkeit des Winkels fr die Parameter des Bildes 5 und (a) = 25, c = 0; (b) = 35, c = 0 kPa; (c) = 25, c = 20 kPa

40



sary support pressure as a function of the angle for thethree sets of shear strength parameters of Figure 5. Thethick solid curves were calculated for = 0.8 using themethod of slices (Equation 32), while the other curveswere obtained assuming the distribution of equation (2)with = 0.8 for the prism and different values w of thiscoefficient for the wedge. Making the same assumption asin the method of slices (i.e. w = = 0.8), the simplifiedmodel leads to a lower support pressure. In order to ob-tain the same frictional resistance (and consequently thesame support pressure), the simplified model of equation(2) should be applied in combination with a lower coeffi-cient w at the lateral wedge planes. In fact, Figure 6ashows that when reducing the w-value according to theratio of the average vertical stresses of Figure 5 (i.e., takingw = z,av/z,av,lin = 0.8 x 112/132 = 0.68) the simplifiedmodel agrees well with the method of slices.

Similar remarks apply to the case of a higher frictionangle (Figure 6b) or of a cohesive ground (Figure 6c), themain difference being that horizontal arching is more pro-nounced in these cases and consequently the differencebetween the two models is bigger. In the case of = 35(Figure6b), the simplified model predicts about the samesupport pressure if the w-value is taken to be 0.55. The re-duction factor w/ = 0.55/0.80 = 0.69 agrees well with theratio of the average stresses (z,av/z,av,lin = 74/110 = 0.67according to Figure 5). This is true also for the cohesiveground (according to Figures 5 and 6c, z,av/z,av,lin =58/96 = 0.60 and w/ = 0.49/0.80 = 0.61, respectively).

Figure 7a shows the results of a parametric studyconcerning the ratio of the average vertical stresses of thetwo models, which at the same time represents the reduc-tion factor to be applied to the w-value of the simplifiedmodel. The diagram shows the stress ratio as a function ofthe cohesion for different values of the friction angle andof the angle . For the parameter combinations of wedgesneeding support (i.e. parameter combinations leading topositive values of support pressure, Figure7b), the reduc-tion factor amounts to 0.50 0.85. Taking w as equal to0.5 (as suggested by Anagnostou and Kovri [1]) there-fore represents a reasonably conservative assumption. Asmentioned above, the same is true with regard to a -valueof 0.8.

4 Comparative calculations concerning support pressure

Figure 8 shows the effect of the depth of cover h on thesupport pressure s. It can readily be seen that, with the ex-ception of soils of very low friction angle, the supportpressure has practically reached its maximum value al-ready at a depth of h = H. In the remaining part of the pre-sent paper all calculations assume that the depth of coveris larger than this (h > H), which in practical terms meansthat the overburden amounts at least to one tunnel diame-ter.

Figure 9 shows the normalized support pressures/D (for the most unfavourable angle ) as a function ofthe normalized cohesion c/D for different friction angles and for = 0.8 or 1. The diagram applies to a circulartunnel face of diameter D. It was calculated by means ofequation (39) considering a quadratic cross section ofequal area (H = B = 0.886 D). Figure 10 compares (for a

specific value of the friction angle) the results obtained us-ing the method of slices with the predictions under themodel of Anagnostou and Kovri [1] and with the resultsof Krause [13] and Vermeer et al. [11]. As mentioned above,the method of Anagnostou and Kovri [1] assumes thesimplified distribution of equation (2) with a reduced lat-eral pressure coefficient for the wedge (w = 0.5 ). The re-sults of Vermeer et al. [11] are based upon three-dimen-

Fig. 8. Normalized support pressure s/H as a function ofthe normalized depth of cover h/H for a granular soil (c = 0)and a circular tunnel (B/H=1)Bild 8. Normierter Sttzdruck s/H in Abhngigkeit der normierten berlagerungshhe h/H fr einen rolligen Boden(c = 0) und einen kreisfrmigen Tunnelquerschnitt (B/H=1)

Fig. 9. Normalized support pressure s/D as a function ofthe normalised cohesion c/D for = 15 35 and = 0.8 or1.0 according to the method of slices (h > H, B/H = 1, D = 2H/ )Bild 9. Normierter Sttzdruck s/D in Abhngigkeit der normierten Kohsion c/D fr = 15 35 und = 0,8 bzw.1,0 nach der Lamellenmethode (h > H, B/H = 1, D = 2H/ )

41



sional numerical stress analyses and can be summarisedas follows:

(55)

where

(56)

(57)

Equation (56) is based upon the results of a comprehen-sive parametric study, while equation (57) is, as mentionedabove, theoretically founded [12] and was also confirmedby the numerical results of Vermeer et al. [11]. Krause [13]investigated a semi-spherical failure mechanism and pro-posed the following coefficients:

(58)

(59)

As observed by Vermeer et al. [11], the method of Anagnos-tou and Kovri [1] leads to slightly higher support pres-sures than the numerical analyses. This is true particularlyfor = 0.8 and to a lesser degree also for = 1.0. The me -thod of slices leads to support pressures which are muchcloser to the numerical predictions of Vermeer et al. [11],and for = 1.0 the difference is irrelevant. These results in-dicate that the reason for the differences from the numeri-cal results is the simplified way of considering horizontalarching in the model of Anagnostou and Kovri [1] [16].

N 1tan

(for 20 , h 2H)c = >

N 19tan

=

N2tanc

=

N 19tan

0.05 (for 20 , h H)= >

sD

N N cDc

=

Due to the linearity of the relationship betweensupport pressure s and cohesion c (Figure9), the resultsof the method of slides can be expressed in terms of only

Fig. 11. Gradient ds/dc as a function of the friction angle according to different computational models (h > H, B/H =1, D = 2H/ ). Remark: The results after Vermeer et al. [11]are practically identical with the results after the method ofslides for = 1.0Bild 11. Gradient ds/dc in Abhngigkeit des Reibungs -winkels nach verschiedenen Berechnungsmethoden (h > H, B/H = 1, D = 2H/ ). Bemerkung: Die Ergebnissenach Vermeer et al. [11] sind praktisch identisch mit den Ergebnissen nach der Lamellenmethode fr = 1.0

Fig. 12. Normalized support pressure s/D of a granular material (c = 0) as a function of the friction angle accor-ding to different computational models (h > H, B/H = 1, D = 2H/ )Bild 12. Normierter Sttzdruck s/D fr einen rolligen Boden (c = 0) in Abhngigkeit des Reibungswinkels nachverschiedenen Berechnungsmethoden (h > H, B/H = 1, D = 2H/ )

Fig. 10. Normalized support pressure s/D as a function of the normalised cohesion c/D for = 25 according to different methods (h > H, B/H = 1, D = 2H/ )Bild 10. Normierter Sttzdruck s/D in Abhngigkeit dernormierten Kohsion c/D fr = 25 nach verschiedenenBerechnungsmethoden (h > H, B/H = 1, D = 2H/ )

42



two parameters the normalised support pressure of acohesionless soil and the gradient of the s(c) line (cf.Equation 3 in [16] as well as Eqs. 39 and 55). Figures 11and 12 show these parameters in the function of the fric-tion angle and compare the different models. The re-sults obtained by the method of slices with = 1.0 agreevery well with the numerical results over the entire para-meter range. The gradient ds/dc is exactly equal to cot(Equation 42).

5 Comparison with experimental data

The computational predictions of the method of slides al-so agree very well with published results of small-scalecentrifuge- [2] [17] [18] or 1g-model tests [19] [20] for tun-nels in cohesionless sand. Figure 13 shows the part of Fig-ure 12 for which test data are available ( -range of 30 to42). The marked rectangles show the range of experimen-tal values. The thick solid line was obtained using themethod of slides. The lines according to Vermeer et al. [11]

and Krause [13] were calculated with equations (62) and(64), respectively. The computational results using themodels of Leca and Dormieux [21] and Kolymbas [22]have been obtained from Kirsch [19].

6 Shape of tunnel cross-section

The enormous influence of horizontal arching can best beillustrated by plotting the necessary support pressure overthe width B of the tunnel face. Figure 14 shows that thenarrower the face, the lower will be the necessary supportpressure. Horizontal arching and the contribution of later-al shear resistance are more pronounced if the face is nar-row. As indicated by the lower curves of Figure 14, a re-duction in width (by partial excavation and vertical subdi-vision of the tunnel cross section; see inset of Figure14)may be sufficient for stabilizing the face, provided that theground exhibits some cohesion. In terms of stability, theeffect of reducing width is therefore similar to that of re-ducing the height of the tunnel face. Moreover, compara-tive calculations show that if the cross section area is keptconstant, the ratio B/H has little influence on the neces-sary support pressure: Figure 15 shows the support pres-sure s (normalized by the diameter D of a circle having thesame area as the face) as a function of the friction anglefor a cohesionless ground (as mentioned above, the effectof cohesion on support pressure is given by ds/dc=cot).The curves apply to markedly different width to height ra-tios B/H but are nevertheless very close together. Conse-quently, the square tunnel cross-section model is reason-ably precise for practical purposes, even for non-circulartunnel cross sections.

Fig. 13. Normalized support pressure s/D of a granular material (c = 0) as a function of the friction angle (part of Figure12): Comparison of the method of slides with experimental data and other computational models (h > H, B/H = 1, D = 2H/ )Bild 13. Normierter Sttzdruck s/D fr einen rolligen Boden (c = 0) in Abhngigkeit des Reibungswinkels (Ausschnitt des Bildes 12): Vergleich der Lamellenmethodemit Versuchsergebnissen und anderen Berechnungs -methoden (h > H, B/H = 1, D = 2H/ )

Fig. 14. Normalized support pressure s/H as a function ofthe normalized width B/H (c = 0, = 25, h > H)Bild 14. Normierter Sttzdruck s/H in Abhngigkeit dernormierten Tunnebreite B/H (c = 0, = 25, h > H)

43



7 Design equation

In conclusion, the method of slices when applied with =1.0 (Terzaghis initial assumption) leads to predictions thatagree well with numerical and experimental results. It canbe verified readily that, for = 1.0 and h > H (a tunnel thatis not too shallow), the coefficient f51 using the method ofslices (Equation 40) can be approximated with sufficientaccuracy by the following equation:

(60)

Inserting equations (60) and (42) into (39) leads to a sim-ple formula, which can be used for estimating the supportpressure:

(61)

If the tunnel cross-section is non-circular, equation(61) can be applied by considering the equivalent diam-eter

(62)

where AT denotes the cross-sectional area of the tunnel.

8 Conclusions

The safety against failure of the 3D mechanism under con-sideration (the wedge and prism model) depends essential-ly on the frictional resistance at the lateral shear plane ofthe wedge and thus on the horizontal stresses. Followingsilo theory, the horizontal stresses can be handled as aconstant percentage of the respective vertical stresses.The simplified model suggested by [6] necessitates, howev-er, an additional assumption concerning the vertical stressz. Comparative calculations show that this model leads

f sD

0.05cot511.75=

s 0.05( cot ) D cot c1.75=

D 2 A /T=

to results similar to those of the method of slices or nu-merical analyses, provided that it is applied in combina-tion with a lower coefficient of lateral stress w. The as-sumptions of = 0.8 and w = 0.4 (suggested in [1] and un-derlying the nomograms [16]) are reasonably conservative.

The method of slices does not require an assumptionconcerning the vertical stress z because the latter resultsfrom the equilibrium equations of the infinitesimal slicesin exactly the same way as in silo theory. For = 1.0, i.e.the value suggested in Terzaghi and Jelinek [15], themethod of slices leads to results that are almost identicalto those of spatial stress analyses, confirming the numeri-cal and theoretical predictions of Vermeer and Ruse [10]regarding the effects of cohesion and tunnel shape, and al-so agreeing well with the experimental data.

References

[1] Anagnostou, G., Kovri, K.: The face stability of slurry-shielddriven tunnels. Tunnelling and Underground Space Technolo-gy 9 (1994), No. 2, pp. 165174.

[2] Idinger, G., Aklik, P., Wu, W., Borja, R.I.: Centrifuge modeltest on the face stability of shallow tunnel. Acta Geotechnica 6(2011), pp. 105117.

[3] Mollon, G., Dias, D., Soubra, A.-H.: Rotational failure me-chanisms for the face stability analysis of tunnels driven by apressurized shield. Int. J. Numer. Anal. Meth. Geomech. 35(2010), pp. 13631388.

[4] Perazzelli, P., Anagnostou, G.: Comparing the limit equilibri-um method and the numerical stress analysis method of tunnelface stability assessment. In 7th Int. Symp. on GeotechnicalAspects of Underground Construction in Soft Ground. Rome,2011.

[5] Walz, B., Pulsfort, M.: Rechnerische Standsicherheit suspen-sionsgesttzter Erdwnde, Teil 1. Tiefbau, Ingenieurbau, Stra-enbau (1983), No. 1, pp. 47.

[6] DIN 4126. Ortbeton-Schlitzwnde; Konstruktion und Aus-fhrung. Berlin: Beuth, 1986.

[7] Broere, W.: Tunnel Face Stability and New CPT Applications.PhD thesis, Delft University of Technology, 2001.

[8] Janssen, H.A.: Versuche ber Getreidedruck in Silozellen.Zeitschrift des Vereines Deutscher Ingenieure (1895), p. 1045.

[9] Walz, B., Prager, J.: Der Nachweis der usseren Standsicher-heit suspensionsgesttzter Erdwnde nach der Elementschei-bentheorie. Verffentlichung des Grundbauinstituts der TUBerlin, 4, 1978.

[10] Vermeer, P.A., Ruse, N.: Die Stabilitt der Tunnelortsbrustin homogenem Baugrund. geotechnik 24 (2001), No. 3, pp.186193.

[11] Vermeer, P.A., Ruse, N., Marcher, Th.: Tunnel Heading Sta-bility in Drained Ground. Felsbau 20 (2002), No. 6, pp. 818.

[12] Ruse, N.: Rumliche Betrachtung der Standsicherheit derOrtsbrust beim Tunnelvortrieb. Universitt Stuttgart, Institutfr Geotechnik, Mitteilung 51, 2004.

[13] Krause, T.: Schildvortrieb mit flssigkeits- und erdgesttzterOrtsbrust. Mitt. des Instituts f. Grundbau und Bodenmechanikder TU Braunschweig, 1987.

[14] Melix, P.: Modellversuche und Berechnungen zur Standsi-cherheit oberflchennaher Tunnel. Verffentlichung des Insti-tuts fr Boden und Felsmechanik der Universitt Fridericianain Karlsruhe, 103, 1987.

[15] Terzaghi, K., Jelinek, R.: Theoretische Bodenmechanik. Ber-lin: Springer-Verlag, 1954.

[16] Anagnostou, G., Kovri, K.: Face stability conditions withEarth Pressure Balanced shields. Tunnelling and UndergroundSpace Technology 11 (1996), No. 2, pp. 165173.

Fig. 15. Normalized support pressure s/D of a granular material (c = 0) as a function of the friction angle for different values of the normalized width B/H (h > H, D = 2H/ )Bild 15. Normierter Sttzdruck s/D fr einen rolligen Boden (c = 0) in Abhngigkeit des Reibungwinkels fr verschiedene Werte der normierten Bandbreite B/H (h > H, D = 2H/ )

44



[17] Chambon P., Corte J.F.: Shallow tunnels in cohesionlesssoil: stability of tunnel face. J Geotech Eng 120 (1994), No 7,pp. 11481165.

[18] Plekkenpol J.W., van der Schrier J.S., Hergarden H.J.: Shieldtunnelling in saturated sandface support pressure and soil de-formations. In Tunnelling: a decade of progress. GeoDelft, 2006.

[19] Kirsch, A.: Experimental investigation of the face stability ofshallow tunnels in sand. Acta Geotechnica 5 (2010), pp. 4362.

[20] Messerli, J., Pimentel, E., Anagnostou, G.: Experimental stu-dy into tunnel face collapse in sand. In Physical Modelling inGeotechnics, Vol. 1, pp. 575580. Zurich, 2010.

[21] Leca, E., Dormieux, L.: Upper and lower bound solutionsfor the face stability of shallow circular tunnels in frictional ma-terial. Geotechnique 40 (1990), No. 4, pp. 581606.

[22] Kolymbas, D.: Tunnelling and Tunnel Mechanics. Berlin:Springer, 2005.

AuthorProf. Dr. sc. techn. Georgios AnagnostouProfessur fr UntertagbauETH Zrich8093 [email protected]

Submitted for review: 9. November 2011Accepted for publication: 25. January 2012

the contribution of horizontal arching to tunnel face stability 2012 geotechnik

Documents

ofthe tunnel face

face figure1b

stability problem

shear stressess

failure model

analternative model

effects of shear resistance

direction ofthe shear